10.3 Hyperbolas10.3 Hyperbolas

Circle Ellipse

Parabola Hyperbola

Conic Sections

See video!

Where do hyperbolas occur?

Hyperbolas

Hyperbola: set of all points such that the difference of the distances from any point to the foci is constant.

Difference of the distances: d2 – d1 = constant

vertices

The transverse axis is the line segment joining the vertices.

The midpoint of the transverse axis is the center of the hyperbola..

asymptotes

d1

d1d2

d2

Standard Equation of a Hyperbola (Center at Origin)

This is the equationif the transverse axis is horizontal.

(–a, 0) (a, 0)

(0, b)

(0, –b)

2 2

2 21

x y

a b

Standard Equation of a Hyperbola (Center at Origin)

This is the equationif the transverse axis is vertical.

(0, –a)

(0, a) (b, 0) (–b, 0)

2 2

2 21

y x

a b

How do you graph a hyperbola?To graph a hyperbola, you need to know the center, the vertices, the fundamental rectangle, and the asymptotes.

Draw a rectangle using +a and +b as the sides...

(–4,0) (4, 0) (0, 3)

(0,-3)

a = 4 b = 3

The asymptotes intersect at the center of the hyperbola and pass through the corners of the fundamental rectangle

Example: Graph the hyperbola

Draw the asymptotes (diagonals of rectangle)...Draw the hyperbola...

2 2

116 9

x y

Example: Write the equation in standard form of 4x2 – 16y2 = 64. Find the vertices and then graph the hyperbola.

Get the equation in standard form (make it equal to 1):

4x2 – 16y2 = 64 64 64 64

(–4,0) (4, 0)

(0, 2)

(0,-2)

That means a = 4 b = 2 Vertices: 4 0 4 0

0 2 0 2

and

( , ), ( , )

( , ), ( , )

Simplify...

2 2

116 4

x y

a

b

y

x

(h, k)

y

x

ab(h, k)

(x – h)2

a2 – (y – k)2

b2 = 1 (y – k)2

a2 – (x – h)2

b2 = 1

Center (h, k) Center (h, k)

Standard Equations for Translated Hyperbolas